reserve x,X,X2,Y,Y2 for set;
reserve GX for non empty TopSpace;
reserve A2,B2 for Subset of GX;
reserve B for Subset of GX;

theorem Th6:
  for V,C being Subset of GX st V is connected & C is connected
  holds Component_of V c= C implies C = Component_of V
proof
  let V,C be Subset of GX;
  assume that
A1: V is connected and
A2: C is connected;
  assume
A3: Component_of V c= C;
  consider F being Subset-Family of GX such that
A4: for A being Subset of GX holds (A in F iff A is connected & V c= A) and
A5: Component_of V = union F by Def1;
  V c= Component_of V by A1,Th1;
  then V c= C by A3;
  then C in F by A2,A4;
  then C c= Component_of V by A5,ZFMISC_1:74;
  hence thesis by A3;
end;
